1,382 research outputs found
Cyclic proof systems for modal fixpoint logics
This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one
Three Essays on Substructural Approaches to Semantic Paradoxes
This thesis consists of three papers on substructural approaches to semantic paradoxes. The first paper introduces a formal system, based on a nontransitive substructural logic, which has exactly the valid and antivalid inferences of classical logic at every level of (meta)inference, but which I argue is still not classical logic. In the second essay, I introduce infinite-premise versions of several semantic paradoxes, and show that noncontractive substructural approaches do not solve these paradoxes. In the third essay, I introduce an infinite metainferential hierarchy of validity curry paradoxes, and argue that providing a uniform solution to the paradoxes in this hierarchy makes substructural approaches less appealing. Together, the three essays in this thesis illustrate a problem for substructural approaches: substructural logics simply do not do everything that we need a logic to do, and so cannot solve semantic paradoxes in every context in which they appear. A new strategy, with a broader conception of what constitutes a uniform solution, is needed
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Certificates for decision problems in temporal logic using context-based tableaux and sequent calculi.
115 p.Esta tesis trata de resolver problemas de Satisfactibilidad y Model Checking, aportando certificados del resultado. En ella, se trabaja con tres lĂłgicas temporales: Propositional Linear Temporal Logic (PLTL), Computation Tree Logic (CTL) y Extended Computation Tree Logic (ECTL). Primero se presenta el trabajo realizado sobre Certified Satisfiability. AhĂ se muestra una adaptaciĂłn del ya existente mĂ©todo dual de tableaux y secuentes basados en contexto para satisfactibilidad de fĂłrmulas PLTL en Negation Normal Form. Se ha trabajado la generaciĂłn de certificados en el caso en el que las fĂłrmulas son insactisfactibles. Por Ășltimo, se aporta una prueba de soundness del mĂ©todo. Segundo, se ha optimizado con Sat Solvers el mĂ©todo de Certified Satisfiability para el contexto de Certified Model Checking. Se aportan varios ejemplos de sistemas y propiedades. Tercero, se ha creado un nuevo mĂ©todo dual de tableaux y secuentes basados en contexto para realizar Certified Satisfiability para fĂłrmulas CTL yECTL. Se presenta el mĂ©todo y un algoritmo que genera tanto el modelo en el caso de que las fĂłrmulas son satisfactibles como la prueba en el caso en que no lo sean. Por Ășltimo, se presenta una implementaciĂłn del mĂ©todo para CTL y una experimentaciĂłn comparando el mĂ©todo propuesto con otro mĂ©todo de similares caracterĂsticas
Fuzzy Natural Logic in IFSA-EUSFLAT 2021
The present book contains five papers accepted and published in the Special Issue, âFuzzy Natural Logic in IFSA-EUSFLAT 2021â, of the journal Mathematics (MDPI). These papers are extended versions of the contributions presented in the conference âThe 19th World Congress of the International Fuzzy Systems Association and the 12th Conference of the European Society for Fuzzy Logic and Technology jointly with the AGOP, IJCRS, and FQAS conferencesâ, which took place in Bratislava (Slovakia) from September 19 to September 24, 2021. Fuzzy Natural Logic (FNL) is a system of mathematical fuzzy logic theories that enables us to model natural language terms and rules while accounting for their inherent vagueness and allows us to reason and argue using the tools developed in them. FNL includes, among others, the theory of evaluative linguistic expressions (e.g., small, very large, etc.), the theory of fuzzy and intermediate quantifiers (e.g., most, few, many, etc.), and the theory of fuzzy/linguistic IFâTHEN rules and logical inference. The papers in this Special Issue use the various aspects and concepts of FNL mentioned above and apply them to a wide range of problems both theoretically and practically oriented. This book will be of interest for researchers working in the areas of fuzzy logic, applied linguistics, generalized quantifiers, and their applications
A Semantic Framework for Neural-Symbolic Computing
Two approaches to AI, neural networks and symbolic systems, have been proven
very successful for an array of AI problems. However, neither has been able to
achieve the general reasoning ability required for human-like intelligence. It
has been argued that this is due to inherent weaknesses in each approach.
Luckily, these weaknesses appear to be complementary, with symbolic systems
being adept at the kinds of things neural networks have trouble with and
vice-versa. The field of neural-symbolic AI attempts to exploit this asymmetry
by combining neural networks and symbolic AI into integrated systems. Often
this has been done by encoding symbolic knowledge into neural networks.
Unfortunately, although many different methods for this have been proposed,
there is no common definition of an encoding to compare them. We seek to
rectify this problem by introducing a semantic framework for neural-symbolic
AI, which is then shown to be general enough to account for a large family of
neural-symbolic systems. We provide a number of examples and proofs of the
application of the framework to the neural encoding of various forms of
knowledge representation and neural network. These, at first sight disparate
approaches, are all shown to fall within the framework's formal definition of
what we call semantic encoding for neural-symbolic AI
A new calculus for intuitionistic Strong L\"ob logic: strong termination and cut-elimination, formalised
We provide a new sequent calculus that enjoys syntactic cut-elimination and
strongly terminating backward proof search for the intuitionistic Strong L\"ob
logic , an intuitionistic modal logic with a provability
interpretation. A novel measure on sequents is used to prove both the
termination of the naive backward proof search strategy, and the admissibility
of cut in a syntactic and direct way, leading to a straightforward
cut-elimination procedure. All proofs have been formalised in the interactive
theorem prover Coq.Comment: 21-page conference paper + 4-page appendix with proof
Natural type inference
Recently, dynamic language users have started to recognize the value of types in their code. To fulfil this need, many popular dynamic languages have adopted extensions that support type annotations. A prominent example is that of TypeScript which offers a module system, classes, interfaces, and an optional type system on top of JavaScript.
However, providing usable (not too verbose, or complex) types via traditional type inference is more challenging in optional type systems. Motivated by this, we redefine the goal of type inference for optionally typed languages as: infer the maximally natural and sound type, instead of the most general one. By the maximally natural and sound, we refer to a type that (1) is derivable in the type system, and (2) maximally reflects the intention of the programmer with respect to a learnt model.
We formally devise a type inference problem that aids the inference of the maximally natural type. Towards this goal, our problem asks to combine information derived from two sources: (1) from algorithmic type systems using deductive logic-based techniques; and (2) from the source code text using inductive machine learning techniques.
To tackle our formulated problem, we develop two frameworks that combine the two sources of information using mathematical optimization. In the first framework, we formulate the inference problem as a problem in numerical optimization. In the second framework, we map the inference problem into popular problems in discrete optimization: maximum satisfiability (MaxSAT) and Integer Linear Programming (ILP).
Both frameworks are built to be consistent with information derived from the different sources. Moreover, through formal proofs, we validate the soundness and completeness of the developed framework for a core lambda-calculus with named types.
To assess the efficacy of the developed frameworks, we implement them in a tool named Optyper that realizes natural type inference for TypeScript. We evaluate Optyperon TypeSript programs obtained from real world projects. By evaluating our theoretical frameworks we show that, in practice, the combination of logical and natural constraints yields a large improvement in performance over either kind of information individually. Further, we demonstrate that our frameworks out-perform state-of-the-art techniques in type inference to produce natural and sound types
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